Revisitando a geometria das horoesferas do espaço hiperbólico.

Abstract

In this work, we revisit the problem of studying the geometry of horospheres of the hyperbolic space Hn+1, with the purpose of characterizing them under certain appropriate constraints of their higher order mean curvatures. In particular, we obtain a gap type result concerning the scalar curvature of complete two-sided hypersurfaces immersed in Hn+1. Furthermore, we establish a estimate for the index of minimum relative nullity of r-minimal (2 ≤ r ≤ n − 1) hypersurfaces of Hn+1 and we also get a nonexistence result for 1-minimal hypersurfaces in the closed horoball determined by a horosphere of Hn+1. Our approach is based on a suitable version of the generalized maximum principle of Omori–Yau for trace-type operators defined on a complete Riemannian manifold with sectional curvature bounded from below.